With the above simplifications and an approximate formulation of the expected cost, we arrive at the following deterministic approximation of the stochastic OCP () in terms of the predicted state mean \(\mathbf {\mu } := \lbrace  \mu ^x_i \rbrace _{i=0}^N\), state covariance \(\mathbf {\Sigma } := \lbrace  \Sigma ^x_i \rbrace _{i=0}^N\) and control input sequence \(\mathbf {u} := \lbrace u_i\rbrace _{i=0}^{N-1}\), cf. [1]},
\(\underset{\mathbf {\mu },\mathbf {\Sigma },\mathbf {u}}{\min } \quad & c_f(\mu _N^x) + \sum _{i=0}^{N-1} c_i(\mu _i^x, u_i) \\\mathrm {s.t.} \quad & \forall i \in \lbrace 0,\ldots ,N-1\rbrace , \\& \mu _{i+1}^x = \psi (\mu _i^x, u_i) + B \mu ^d(\mu _i^x, u_i),  \\& \Sigma _{i+1}^x = \tilde{A}_i \Sigma _i^x \tilde{A}_i^\top + B \left( \Sigma ^d(\mu _i^x, u_i) + \Sigma ^w \right) B^\top , \\ & \bar{h}_j(\mu ^x_i,u_i,\Sigma ^x_i) \le 0, \: j=1,\ldots ,n_h, \\& \mu _0^x = x(k) , \\& \Sigma _0^x = 0 .\) 
